Stationary measures for the Kardar-Parisi-Zhang equation

The Kardar-Parisi-Zhang (KPZ) equation is a nonlinear stochastic partial differential equation introduced in physics in 1986. In one spatial dimension, for the KPZ equation on the line, it has been known for a long time that the Brownian measure is stationary. For the equation on an interval or a half-line, however, stationary measures are more complex and their explicit description have been obtained very recently. The method involves the detailed study of integrable probabilistic models that can be viewed as discretizations of the KPZ equation. The talk is based on joint works with Pierre Le Doussal and Ivan Corwin.